# Order of a m-cycle

• Oct 15th 2011, 08:21 PM
Bernhard
Order of a m-cycle
Dummit and Foote Section 1.3 Symmetric Groups - Exercise 10 states:

Prove that if $\displaystyle \sigma$ is the m-cycle ($\displaystyle a_1$ $\displaystyle a_2$ ... $\displaystyle a_m$) then for all i $\displaystyle \in$ {1,2, ... m} we have $\displaystyle {\sigma}^i$($\displaystyle a_k$) = $\displaystyle a_{k+i}$ where k+i is replaced by its least positive residue mod m. Deduce that |$\displaystyle \sigma$| = m.

Can anyone help with this problem? Would be grateful for help!

I think I can see how this works but I am struggling with how to write a clear and valid formal proof.

Peter
• Oct 15th 2011, 08:43 PM
alexmahone
Re: Order of a m-cycle
Quote:

Originally Posted by Bernhard
Dummit and Foote Section 1.3 Symmetric Groups - Exercise 10 states:

Prove that if $\displaystyle \sigma$ is the m-cycle ($\displaystyle a_1$ $\displaystyle a_2$ ... $\displaystyle a_m$) then for all i $\displaystyle \in$ {1,2, ... m} we have $\displaystyle {\sigma}^i$($\displaystyle a_k$) = $\displaystyle a_{k+i}$ where k+i is replaced by its least positive residue mod m. Deduce that |$\displaystyle \sigma$| = m.

Can anyone help with this problem? Would be grateful for help!

I think I can see how this works but I am struggling with how to write a clear and valid formal proof.

Peter

Note that $\displaystyle \sigma^m=e$ and $\displaystyle \sigma^n\neq e$ for $\displaystyle n$ ranging from $\displaystyle 1$ to $\displaystyle m-1$.
• Oct 15th 2011, 08:52 PM
Bernhard
Re: Order of a m-cycle
Thanks

Yes, can see that! Would that constitute a satisfactory proof?

Peter
• Oct 15th 2011, 08:54 PM
alexmahone
Re: Order of a m-cycle
Quote:

Originally Posted by Bernhard
Would that constitute a satisfactory proof?

Depends on whether you have a satisfactory argument for $\displaystyle \sigma^n\neq e$ for $\displaystyle n$ ranging from $\displaystyle 1$ to $\displaystyle m-1$.
• Oct 15th 2011, 09:01 PM
Deveno
Re: Order of a m-cycle
if you prove the first part, you'll have a satisfactory proof of the second part.